Stochastic models of tumor growth and the probability of elimination by cytotoxic cells

The probability of tumor extinction due to the action of cytotoxic cell populations is investigated by several one dimensional stochastic models of the population growth and elimination processes of a tumor. The several models are made necessary by the nonlinearity of the processes and the different parameter ranges explored. The deterministic form of the model is where γ₀, k′₆ and k ₁ are positive constants. The parameter of most import is which determines the stability of the T = 0 equilibrium. With an initial tumor size of one, a (linear) branching process is used to estimate the extinction probability. However, in the case λ = 0 when the linearization of the deterministic model gives no information (T = 0 is actually unstable) the branching model is unsatisfactory. This makes necessary the utilization of a density-dependent branching process to approximate the population. Through scaling a diffusion limit is reached which enables one to again compute the probability of extinction. For populations away from one a sequence of density-dependent jump Markov processes are approximated by a sequence of diffusion processes. In limiting cases, the estimates of extinction correspond to that computed from the original branching process. Table 1 summarizes the results.     

Morphological and random amplified polymorphic DNA features in populations ofCeramium kondoi (Rhodophyta)

Morphological and RAPD features ofCeramium kondoi populations were investigated and compared in different locations in Korea. The plant length and branching pattern were more variable in Jindo population than others. RAPD data showed thatC. kondoi plants were divided into two clades; the southern group including Jindo and Bangpo population, and the northern group including Yonpyongdo and Oeyondo population. Morphological features inC. kondoi populations corresponded with RAPD data, which differed from those ofC. boydenii from the same location. These results suggest that RAPD might be useful for elucidating genetic variation among the wild populations ofC. kondoi.     

MORPHOLOGICAL AND GENETIC VARIATION IN THE POPULATIONS OF SARGASSUM HEMIPHYLLUM (PHAEOPHYCEAE) IN THE NORTHWESTERN PACIFIC1

Difficulty in species identification of Sargassum (Sargassaceae, Fucales) is partly attributed to the high polymorphism among its individuals and populations. This study aimed at assessing morphological and genetic variations in two varieties, var. hemiphyllum J. Agardh and var. chinense J. Agardh, of Sargassum hemiphyllum (Turner) C. Agardh, a widely distributed species in the northwestern Pacific. We investigated 26 measurable, five numerical, and 33 categorical morphological parameters associated with different branching levels of specimens from each of six localities within its distribution range using cluster analysis (CA) and principal coordinate analysis (PCoA). Leaf size of the primary and secondary branching levels and the vesicle size of the secondary branches of the specimens examined were determined to be the most important morphological parameters that were significantly different among populations. Change in leaf and vesicle length of individuals among the six populations followed a latitudinal gradient, with smaller leaves and vesicles associated with northern populations and larger ones in the southern populations. The possible influence of the gradual change in sea surface temperatures (SSTs) along this gradient in the northwestern Pacific on leaf and vesicle morphologies of this species was suggested. PCR‐RFLP analysis of the RUBISCO spacer in the chloroplast genome revealed two distinct and highly homogenous clades, a China clade and a Japan‐Korea clade, which corresponded to var. chinense and var. hemiphyllum, respectively. The formation of refugia along the “Paleo‐coast” in the East China Sea during glacial periods is suggested to have led to the vicariance of ancestral populations of S. hemiphyllum and thus to have promoted genetic differentiation. The massive freshwater outflow of the Yellow and Yangtze rivers may continue to act as a barrier, prolonging the allopatric distribution of the two varieties.     

Pleiotropy of the branching locus (B) masks linked and unlinked quantitative trait loci affecting seed traits in sunflower

The discovery of unbranched, monocephalic natural variants was pivotal for the domestication of sunflower (Helianthus annuus L.). The branching locus (B), one of several loci apparently targeted by aboriginal selection for monocephaly, pleiotropically affects plant, seed and capitula morphology and, when segregating, confounds the discovery of favorable alleles for seed yield and other traits. The present study was undertaken to gain deeper insights into the genetics of branching and seed traits affected by branching. We produced an unbranched hybrid testcross recombinant inbred line (TC-RIL) population by crossing branched (bb) and unbranched (BB) RILs to an unbranched (BB) tester. The elimination of branching concomitantly eliminated a cluster of B-linked seed trait quantitative trait loci (QTL) identified by RIL per se testing. We identified a seed oil content QTL linked in repulsion and a 100-seed weight QTL linked in coupling to the B locus and additional unlinked QTL, previously masked by B-locus pleiotropy. Genomic segments flanking the B locus harbor multiple loci for domestication and post-domestication traits, the effects of which are masked by B-locus pleiotropy in populations segregating for branching and can only be disentangled by genetic analyses in unbranched populations. QTL analyses of NILs carrying wild B alleles substantiated the pleiotropic effects of the B locus. The effect of the B locus on branching was masked by the effects of wild alleles at independent branching loci in hybrids between monocephalic domesticated lines and polycephalic wild ecotypes; hence, the B locus appears to be necessary, but not sufficient, for monocephaly in domesticated sunflower.     

The effective founder effect in a spatially expanding population

The gradual loss of diversity and the establishment of clines in allele frequencies associated with range expansions are patterns observed in many species, including humans. These patterns can result from a series of founder events occurring as populations colonize previously unoccupied areas. We develop a model of an expanding population and, using a branching process approximation, show that spatial gradients reflect different amounts of genetic drift experienced by different subpopulations. We then use this model to measure the net average strength of the founder effect, and we demonstrate that the predictions from the branching process model fit simulation results well. We further show that estimates of the effective founder size are robust to potential confounding factors such as migration between subpopulations. We apply our method to data from Arabidopsis thaliana. We find that the average founder effect is approximately three times larger in the Americas than in Europe, possibly indicating that a more recent, rapid expansion occurred.     

Adaptive branching in source-sink habitats

Evolution and ecological diversification in a heterogeneous environment is driven by an often complex interplay between local adaptation and dispersal between different habitat types. Heterogeneous environments also easily generate source-sink dynamics of populations coupled by dispersal. It follows that local adaptation and possible adaptive radiation almost by necessity involves adaptation to a (pseudo-)sink habitat, which is considered unlikely. We here study a model of ‘parapatric branching’ with this special focus on the spatial ecology of the process. We find that evolutionary branching can display a sequence of alternating adaptations to the source or the sink. In some circumstances a true sink can become a pseudo-sink through adaptation to the corresponding source habitat. The evolutionary endpoint is a spatially structured community consisting of two source populations with one corresponding sink or pseudo-sink each. Our results shed new light on the interpretation of extant source-sink systems and the process of parapatric branching.     

The rate of convergence of a generalized stable population

In an age-structured population that grows exponentially, each age groupP _i(t) at periodt is asymptotically equivalent tox ₀ ^t for some positive number x₀. In this paper we show that the speed at which the ith age group reaches its exponential state of equilibrium can be measured by the rate at which the ratio v_i(t)=P_i(t)/p_i(t–1) converges tox ₀. The age specific rate of convergence is determined by considering a quantityr satisfyingv _i(t)-x ₀ ¦ r ^t whent is large;R _i=Infr (over all initial populations,r satisfying the above inequality) is the R-factor used in numerical analysis to measure the rate at which the sequencev _i (t) converges tox ₀;S _i =- In R_i is then defined as the rate of convergence to stability of the ith age group. The case of constant net maternity rates is studied in detail; in this contextS ₀ is compared to the population entropyH, which was proposed by Tuljapurkar (1982) as a measure of the rate of convergence to stability.     

A branching process model for the evolution of transposable elements

A discrete-time multitype branching process model is presented for the evolution of transposable elements in haploid populations. An individual is classified as type i if it possesses i copies of the TE, i0. The general model incorporates copy-dependent selection and transposition, and recursion relations are derived for the distribution of the number of individuals of the various types. The asymptotic relative proportions of individuals of the different types is studied in the neutral case. The behavior of this equilibrium distribution is examined for various patterns of regulated transposition and deletion.     

The evolution of resource use

 The evolution of a consumer exploiting two resources is investigated. The strategy x under selection represents the fraction of time or energy an individual invests into extracting the first resource. In the model, a dimensionless parameter α quantifies how simultaneous consumption of both resources influences consumer growth; α<0 corresponds to hemi-essential resources, 0<α<1 corresponds to complementary resources, α=1 corresponds to perfectly substitutable resources, and α>1 corresponds to antagonistic resources. An analysis of the ecological and evolutionary dynamics leads to five conclusions. First, when α≤1, there is a unique singular strategy x ^* for the adaptive dynamics and it is evolutionarily stable and globally convergent stable. Second, when α=1, the singular strategy x ^* corresponds to the populations exhibiting an ideal free distribution and a population playing this strategy can invade and displace populations playing any other strategy. Third, when α>1, the strategies x=0 and x=1 are evolutionarily stable and convergent stable. Hence, if the populations initially specialize on one resource, evolution amplifies this specialization. Fourth, when α is slightly larger than one (i.e. the resources are slightly antagonistic), there is a convergent stable singular strategy whose basin of attraction is almost the entire strategy space (0,1). This singular strategy is evolutionarily unstable and serves as an evolutionary branching point. Following evolutionary branching, our analysis and numerical simulations suggest that evolutionary dynamics are driven toward an end state consisting of two populations specializing on different resources. Fifth, when α>>1, there is only one singular strategy and it is convergent unstable and evolutionarily unstable. Hence, if resources are overly antagonistic, evolutionary branching does not occur and ultimately only one resource is exploited. Received: 8 June 2002 / Revised version: 28 November 2002 / Published online: 23 April 2003 This work was supported by NSF Grant DMS-0077986 Key words or phrases: Consumer-resource interactions – Adaptive dynamics – Evolutionary branching     

Detecting particular genotypes in populations under nonrandom mating

We investigate the time to formation of particular genotypes in populations with nonrandom mating systems. We employ two main techniques. The first is a study of branching processes with “killing”; these are models which behave just like a standard Galton-Watson branching process with the added possibility of being terminated by the occurrence of a special event in the process. In our case, this special event corresponds to formation or detection of a group of individuals carrying a specific genotype. We then use these results and some natural approximation methods to analyze and interpret the gene formation problem in a simple way.     

BRANCHING PATTERNS OF SALICORNIA EUROPAEA (CHENOPODIACEAE) AT DIFFERENT SUCCESSIONAL STAGES: A COMPARISON OF THEORETICAL AND REAL PLANTS

Salicornia europaea L. (Chenopodiaceae) is an annual succulent halophyte that lacks leaves, is photosynthetically active over its entire surface, and branches in a predictable manner. A computer model based on that of Niklas and Kerchner (1984) was used to generate S. europaea-like branching patterns. The model was used to predict the morphology of S. europaea that could maximize light interception and minimize the total bending moment. The optimal branching pattern generated by the computer corresponded very closely to the form of S. europaea found in late-successional populations. The progression of model forms from the least efficient (lowest total projected surface area and highest bending moment) to the most efficient (highest projected surface area and lowest bending moment) parallels the observed phenotypic changes in morphology of S. europaea over the course of succession in New England salt marshes. Based on computer simulations, we conclude that morphological changes in the branching patterns of S. europaea during succession correspond to alterations of shape capable of coincidentally maximizing the interception of light and minimizing the total bending moment.     

Developmental morphology of branching flowers in Nymphaea prolifera

Nymphaea and Nuphar (Nymphaeaceae) share an extra-axillary mode of floral inception in the shoot apical meristem (SAM). Some leaf sites along the ontogenetic spiral are occupied by floral primordia lacking a subtending bract. This pattern of flower initiation in leaf sites is repeated inside branching flowers of Nymphaea prolifera (Central and South America). Instead of fertile flowers this species usually produces sterile tuberiferous flowers that act as vegetative propagules. N. prolifera changes the meristem identity from reproductive to vegetative or vice versa repeatedly. Each branching flower first produces some perianth-like leaves, then it switches back to the vegetative meristem identity of the SAM with the formation of foliage leaves and another set of branching flowers. This process is repeated up to three times giving rise to more than 100 vegetative propagules. The developmental morphology of the branching flowers of N. prolifera is described using both microtome sections and scanning electron microscopy.     

Molecular Basis of Unique Branching Phenotypes in <i>Salvia splendens</i> and the Role of PSY

The branching system of higher plants plays a very important role in plant morphogenesis, and the number of branches can directly affect crop yield and the ornamental value of plants. It is a complicated development process involving complex molecular mechanisms. The ‘Cailinghong’ variety of Salvia splendens is characterized by its great branching ability with the ability to grow into a spherical form naturally, without pinching. To gain insight into the molecular events during the branching development of S. splendens, suppressive subtractive hybridization (SSH) technology was used to screen differentially expressed genes between the erect plant type (strain 35) and the spherical plant type (‘Cailinghong’). In total, 96 and 116 unigenes were annotated. Four and eight unigenes up-regulated in ‘Cailinghong’ and strain 35, respectively, were associated with plant hormone anabolism and signal transduction, suggesting that they participate in the branching process. One of these genes, phytoene synthase (PSY), is a precursor of the new plant hormone group strigolactones. Using the PSY fragment (192 bp) as a template, the cDNA sequence of PSY in S. splendens was cloned and named SsPSY. A relative expression analysis and transgenic test results indicated that SsPSY plays an important role in lateral branch development in ‘Cailinghong’. These results provide new insight into the molecular mechanisms underlying branching in S. splendens.     

On a periodic-like behavior of a delayed density-dependent branching process

Most of natural populations seem to be regulated in their sizes in complex ways. Particularly, the sizes of some populations change in time or generation roughly periodically. There are many theoretical studies on such population dynamics. This paper develops stochastic population models for a periodic-like population dynamics. To see the nature of such mechanism, we consider simple models of a delayed density-dependent branching process, and present by numerical simulations how such a branching process shows periodic population changes. The effects of randomly changing stationary environments on the population dynamics are also considered.     

The effective population size of some age-structured populations

It was shown in a previous paper that if generations are discrete, then the effective population size of a large population can be derived from the theory of multitype branching processes. It turns out to be proportional to the reciprocal of a term that appears in the denominator of expressions for survival probabilities when there is a supercritical positively regular branching process for which the dominant positive eigenvalue of the first moment matrix is slightly larger than 1. If there is an age-structured population with unchanging proportions among sexes and age groups, then the effective population size is shown to be also obtainable from the theory of multitype branching processes. The expression for this parameter has the same form as in the corresponding model for discrete generations, multiplied by an appropriate measure of the average length of a generation. Results are obtained for dioecious random mating populations, populations reproducing partly by selfing, and populations reproducing partly by full-sib mating.     

On the study of two interacting populations one of which acts independently of the other

Many ecological and biological systems can be studied in terms of a bivariate stochastic branching process, {X ₁ (t), X ₂ (t)}, each of whose components (or populations) varies in magnitude according to the laws of a generalized birth-death process. Of particular interest is such a model in which the birth and death rates of the first population,X ₁, are constant while those of the second population,X ₂, exhibit a functional dependence upon the magnitude of the first. It is shown, first, that the existence of the stochastic mean of a birth death process implies the existence of all higher moments. The values of all the factorial moments of such a process are then determined. The moments of the dependent population of the bivariate process are given in terms of its expectation and the joint probability density function of the process is determined. It is possible, therefore, to use Bayesian techniques to infer conclusions about the independent population, given information about the variation of the dependent one.     

Correlations between chaetodontid fishes and coral communities of the Gulf of Aqaba (Red Sea)

Synopsis The relationships existing between the chaetodontid fishes and the surrounding coral communities were investigated in the Gulf of Aqaba. Quantitative data were analysed by a correspondence and a cluster analysis. The results demonstrated a similarity in the spatial distribution of both communities. Significant correlations were found between the density of chaetodontid fishes and the diversity of the coral community as well as the substratum coverage by the coral colonies. The density of exclusive coral browsers was also correlated to the abundance of branching colonies. Among the different genera of branching corals, correlations were significant only for the genusAcropora. These results suggested the existence of strong links between coral and chaetodontid fish assemblages.     

Polymorphism pattern at a miniature inverted‐repeat transposable element locus downstream of the domestication gene Teosinte‐branched1 in wild and domesticated pearl millet

Unravelling the mechanisms involved in adaptation to understand plant morphological evolution is a challenging goal. For crop species, identification of molecular causal polymorphisms involved in domestication traits is central to this issue. Pearl millet, a domesticated grass mostly found in semi‐arid areas of Africa and India, is an interesting model to address this topic: the domesticated form shares common derived phenotypes with some other cereals such as a decreased ability to develop basal and axillary branches in comparison with the wild phenotype. Two recent studies have shown that the orthologue of the maize gene Teosinte‐Branched1 in pearl millet (PgTb1) was probably involved in branching evolution during domestication and that a miniature inverted‐repeat transposable element (MITE) of the Tuareg family was inserted in the 3′ untranslated region of PgTb1. For a set of 35 wild and domesticated populations, we compared the polymorphism patterns at this MITE and at microsatellite loci. The Tuareg insertion was nearly absent in the wild populations, whereas a strong longitudinal frequency cline was observed in the domesticated populations. The geographical pattern revealed by neutral microsatellite loci clearly demonstrated that isolation by distance does not account for the existence of this cline. However, comparison of population differentiation at the microsatellite and the MITE loci and analyses of the nucleotide polymorphism pattern in the downstream region of PgTb1 did not show evidence that the cline at the MITE locus has been shaped by selection, suggesting the implication of a neutral process. Alternative hypotheses are discussed.     

Branching out: a remarkable new branching syllid (Annelida) living in a Petrosia sponge (Porifera: Demospongiae)

We describe the morphology and biology of a previously unknown form of branching annelid, Ramisyllis multicaudata gen. et sp. nov. , an endosymbiont of shallow‐water marine sponges (Petrosia sp., Demospongiae) in northern Australia. It belongs to the polychaete family Syllidae, as does Syllis ramosa McIntosh, 1879, the only other named branching annelid, which was collected from deep‐water hexactinellid sponges during the 1875 Challenger expedition. It differs from S. ramosa in parapodial and chaetal morphology. Ramisyllis multicaudata gen. et sp. nov. has segments of several types, including specialized posterior segments on the emergent portions of the worm, and simplified elongate segments that bridge larger cavities in the sponge interior. Aside from the obvious branching form, the new annelid is similar to Parahaplosyllis, differing from it in lacking pharyngeal armature and in the details of the parapodial chaetae and dorsal cirri. Molecular evidence from 16S and 18S rDNA supports a sister‐group relationship with Parahaplosyllis, with both being sister to Trypanosyllis and Eurysyllis. The phylogenetic position of R. multicaudata gen. et sp. nov. indicates that branching has evolved independently in Ramisyllis gen. nov. and Syllis. This is supported by differences in the branching process between the two taxa: in S. ramosa branching is initiated by segment addition at the parapodium, whereas in R. multicaudata gen. et sp. nov. segments are added from a region between parapodia. A model for branching in R. multicaudata gen. et sp. nov. is proposed and possible developmental processes underlying branching in Annelida, and body symmetry comparisons with other invertebrates, are also discussed. © 2012 The Linnean Society of London, Zoological Journal of the Linnean Society, 2012, 164 , 481–497.